Consider two cylindrical rods of identical dimensions, one of rubber and the other of steel. Both the rods are fixed rigidly at one end to the roof. A mass M is attached to each of the free ends at the center of the rods

The correct option is: The steel rod will elongate, without any perceptible change in shape, but the rubber rod will elongate with the shape of the bottom edge tapered to a tip at the center.

Step-by-Step Explanation:

1. Elastic Modulus and Deformability:
Steel has a very high Young's modulus (denoted as $Y$), which means it is highly rigid and resists deformation. Under a load $M$, the strain in the steel rod is extremely small. The steel rod will undergo a uniform vertical elongation, but its cylindrical shape and the flat surface of its bottom edge remain practically unchanged and uniform.

2. Shear Modulus and Shape Changes in Rubber:
In contrast, rubber is a highly elastic material with a much lower Young's modulus and a very low shear modulus (rigidity modulus, denoted as $\eta$). This low shear modulus makes it very susceptible to shear strains and localized shape deformations under concentrated forces.

3. Localized Load Application:
The question specifies that the mass $M$ is attached at the center of the free end (the bottom edge) of each rod.
- For the steel rod, the high rigidity distributes the stress evenly across the cross-section, maintaining a flat bottom surface.
- For the rubber rod, because of its low resistance to shear forces, the concentrated load at the center pulls the central region of the bottom surface downward much more than the outer edges. This localized stretching causes the bottom flat surface of the rubber cylinder to deform into a cone-like shape, tapering to a tip at the center where the load is attached.

4. Conclusion:
Therefore, while the steel rod simply elongates uniformly without any noticeable change in its overall geometry, the rubber rod elongates and undergoes a significant shape distortion, where its bottom edge tapers to a tip at the central point of attachment.